Improving Estimates of Monotone Functions by Rearrangement

TL;DR

This paper demonstrates that rearrangement techniques can effectively improve non-monotonic estimates of monotone functions, making them closer to the true function without loss of generality, supported by computational and empirical examples.

Contribution

It introduces and validates rearrangement methods as a general approach to enhance estimates of monotone functions from non-monotonic estimates.

Findings

Rearrangement improves the accuracy of estimates in monotone function estimation.

Rearranged estimates are closer to the true function in common metrics.

The methods are applicable in both univariate and multivariate settings.

Abstract

Suppose that a target function is monotonic, namely, weakly increasing, and an original estimate of the target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates. We show that these estimates can always be improved with no harm using rearrangement techniques: The rearrangement methods, univariate and multivariate, transform the original estimate to a monotonic estimate, and the resulting estimate is closer to the true curve in common metrics than the original estimate. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.